Consider a rocket that is in deep space and at rest relative to an inertial refe

Question

Consider a rocket that is in deep space and at rest relative to an inertial reference frame. The rocket's engine is to be fired for a certain interval. What must be the rocket's mass ratio (ratio of initial to final mass) over that interval if the rocket's final speed relative to the inertial frame is to be equal to (a) the exhaust speed (speed of the exhaust products relative to the rocket) and (b) 4.00 times the exhaust speed? (The exhaust products from the rocket escape at a constant velocity relative to the rocket)

Explanation / Answer

dv(M - dm) = dm/dt * v.exhaust
dv = v.exhaust/((M -dm)*dt) dm

Ignore dt cause you aren't interested. You can also throw out the "dm" in the denominator (M -dm) as dminimus

dv = v.exhaust/M dm
? v = v.exhaust (ln m.i - ln m.f)

But we know that

? v/v.exhaust = 1
ergo,
ln m.i - ln m.f = 1
m.i/m.f = e

You can do it for 2

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